Musical Tonality

1
Musical Tonality

• Existence of multiple tonalities raises a variety
  of questions
• What is the relation of one tonality to another?
• Can we measure relatedness?
• How do we represent relatedness?
• Questions regarding perception of tonality
• How do we find our sense of key?
• Are we accurate in sense of key?
• What models do we have for determining key?

2
Theoretical Maps of Interkey Distances

• Are there ways of deriving interkey distances?
• Put differently, can we determine key maps?
• Key distance an important issue in
  music-theoretic descriptions of tonal music
  because of key modulation
• Musical composition written in home key
• Compositions move to other keys, or modulate
• Usually return to home key
• Practice codified in interkey distances key
  distances considered close if modulations are
  frequent

3
Interkey Distances
The Circle of Fifths
4
Interkey Distances, cont
Diatonic set overlaps
5
Interkey Distances, cont
Diatonic set overlaps

• The problem of minor keys
• Different forms of the minor

• Relation between the major and minor keys
• The Relative Minor (C major and A minor)
• Identical scale degrees
• C Major C D E F G A B
• A Minor A B C D E F G
• The Parallel Minor (C major and C minor)
• Common tonic and 5th scale degree
• C Major C D E F G A B
• C Minor C D Eb F G Ab B

6
Interkey Distances, cont
Krumhansl Kessler (1982)
7
Interkey Distances, cont
Interkey correlations
8
Interkey Distances, cont
Interkey correlations, graphed
9
Interkey Distances, cont
Krumhansl Kessler (1982) Multidimensional
scaling solution
10
Interkey Distances, cont
The four-dimensional torus
11
Interkey Distances, cont
Krumhansl Kessler (1982) Map of key space
12
Interkey Distances, cont
Schoenberg (1954/1969) Chart of key distance
13
Interkey Distances, cont
Werts (1983)

• Analyzed key progressions between compositions
• Expressed as movement from one key to neighboring
  keys
• Produced a multidimensional scaling solution
  based on number of times keys moved from one to
  another
• Similar to MDS solution just seen

14
Models of Key Finding
Longuet-Higgens Steedman (1971) C
Major Diatonic Scale C D E F G A B Major
Chords C E G F A C G B D E A B
C G F C G D C Major 0 Major A E B 9 4
11 F C G D 5 0 7 2 C Minor 0
Minor B 11 F C G D 5 0 7 2 G D
8 3
15
Models of Key Finding, cont
Longuet-Higgens Steedman (1971) Map of key
space
A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C 9 4 11 6 1 8 3 10 5 0 7 2 9 5 0 7 2 9 4 11 6
1 8 3 10 5 1 8 3 10 5 0 7 2 9 4 11 6 1 9 4 11 6 1
8 3 10 5 0 7 2 9 5 0 7 2 9 4 11 6 1 8 3 10 5 1 8
3 10 5 0 7 2 9 4 11 6 1 9 4 11 6 1 8 3 10 5 0 7 2
9 5 0 7 2 9 4 11 6 1 8 3 10 5 1 8 3 10 5 0 7 2 9 4
11 6 1
16
Models of Key Finding, cont
Longuet-Higgens Steedman (1971)
Notes in Fugue Subject   E F B C D
E D E F G A 4 6 11 1 3 4 3 4 6 8 9   C D E F
G A B 1 3 4 6 8 9 11
17
Models of Key Finding, cont
Longuet-Higgens Steedman (1971) Map of key
space
A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C Possible Keys Note 1 (E) B, E, A, D,
G, C, F
18
Models of Key Finding, cont
Longuet-Higgens Steedman (1971) Map of key
space
A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C Possible Keys Note 1 (E) B, E, A, D,
G, C, F Note 2 (E, F) B, E, A, D, G
19
Models of Key Finding, cont
Longuet-Higgens Steedman (1971) Map of key
space
A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C Possible Keys Note 1 (E) B, E, A,
D, G, C, F Note 2 (E, F) B, E, A, D,
G Note 3 (E, F, B) B, E, A, D, G
20
Models of Key Finding, cont
Longuet-Higgens Steedman (1971) Map of key
space
A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C Possible Keys Note 1 (E) B, E, A,
D, G, C, F Note 2 (E, F) B, E, A, D,
G Note 3 (E, F, B) B, E, A, G Note 4 (E,
F, B, C) B, E, A
21
Models of Key Finding, cont
Longuet-Higgens Steedman (1971) Map of key
space
A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C Possible Keys Note 1 (E) B, E, A,
D, G, C, F Note 2 (E, F) B, E, A, D,
G Note 3 (E, F, B) B, E, A, G Note 4 (E,
F, B, C) B, E, A Note 5 (E, F, B, C, D)
B, E
22
Models of Key Finding, cont
Longuet-Higgens Steedman (1971) Map of key
space
A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C A E B F C G D A F C G D A F C G D A
E B F C G D A F C G D A F C G D A E B F
C Possible Keys Note 1 (E) B, E, A,
D, G, C, F Note 2 (E, F) B, E, A, D,
G Note 3 (E, F, B) B, E, A, G Note 4 (E,
F, B, C) B, E, A Note 5 (E, F, B, C, D)
B, E . . . Note 10 (E, F, B, C, D, G)
B, E Note 11 (E, F, B, C, D, G, A) E
23
Models of Key Finding, cont
Longuet-Higgens Steedman (1971)
Notes in Fugue Subject   C Db C B E
F Bb A Ab G 0 1 0 11 4 5 10 9 8 7   (0 1 4 5 7 8
9 10 11)
24
Models of Key Finding, cont
Krumhansl-Schmuckler Key-Finding Algorithm
(Krumhansl Schmuckler, 1986, Schmuckler
Tomovski, 1995) Schubert, Op. 94, no. 1 Tone
durations
25
Models of Key Finding, cont
Krumhansl-Schmuckler Key-Finding Algorithm
(Krumhansl Schmuckler, 1986) Bach, C Minor
Prelude Tone durations
26
Models of Key Finding, cont
Krumhansl-Schmuckler Key-Finding Algorithm
(Schmuckler Tomovski, 2005)
27
Models of Key Finding, cont
Krumhansl-Schmuckler Key-Finding Algorithm
(Schmuckler Tomovski, 2005)
28
Models of Key Finding, cont
Brown Butler (1981) Intervals of the diatonic
set Major Scale
29
Models of Key Finding, cont
Brown Butler (1981) Trichords and the rare
interval
30
Models of Key Finding, cont
Brown Butler (1981) Intervals of the diatonic
set Minor Scale